Solving Systems of Equations
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In this chapter we consider the classical problem of solving (exactly) a system of algebraic equations over a field F. This problem, along with the related problem of solving single univariate equations, was the fundamental concern of algebra until the beginning of the “modern” era (roughly, in the nineteenth century); it remains today an important, widespread concern in mathematics, science and engineering. Although considerable effort has been devoted to developing methods for numerical solution of equations, the develop- ment of exact methods is also well motivated. Obviously, exact methods avoid the issues of conditioning and stability. Moreover, in the case of nonlinear systems, numerical methods cannot guarantee that all solutions will be found (or prove that none exist). Finally, many systems which arise in practice contain “free” parameters and hence must be solved over non-numerical domains.
KeywordsIntegral Domain Computer Algebra Gaussian Elimination Common Root Univariate Polynomial
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- 3.S. Cabay, “Exact Solution of Linear Equations,” pp. 392–398 in Proc. SYMSAM’ 71, ed. S.R. Petrick, ACM Press (1971).Google Scholar
- 5.J.F. Canny, E. Kaltofen, and L. Yagati, “Solving Systems of Non-Linear Polynomial Equations Faster,” pp. 121–128 in Proc. ISSAC’ 89, ed. G.H. Gonnet, ACM Press (1989).Google Scholar
- 7.G.E. Collins, “Quantifier Elimination for Real Closed Fields: A Guide to the Literature,” pp. 79–81 in Computer Algebra — Symbolic and Algebraic Computation (Second Edition), ed. B. Buchberger, G.E. Collins and R. Loos, Springer-Verlag, Wien — New York (1983).Google Scholar
- 11.D. Lazard, “Systems of Algebraic Equations,” pp. 88–94 in Proc. EUROSAM’ 79, Lecture Notes in Computer Science 72, ed. W. Ng, Springer-Verlag (1979).Google Scholar
- 12.J.D. Lipson, “Symbolic methods for the computer solution of linear equations with applications to flowgraphs,” pp. 233–303 in Proc. of the 1968 Summer Inst. on Symb. Math. Comp., ed. R. G. Tobey, (1969).Google Scholar
- 15.H. Takahasi and Y. Ishibashi, “A New Method for’ Exact Calculation’ by a Digital Computer,” Inf. Processing in Japan, 1 pp. 28–42 (1961).Google Scholar
- 16.B.L. van der Waerden, Modern Algebra (Vols. I and II), Ungar (1970).Google Scholar